How to simulate a sheet of paper in Blender Game Engine?
The biggest problem I see here is that the softbody algorithms in the game engine are not physically correct, they are only made to look plausible and be fast. Especially the stiffness of softbodies depends on the mesh resolution: higher subdivision results in less stiffness.

A setup that looks fine for paper is a plane with just a few subdivisions



and the following Softbody settings



Make sure you disable Shape Match since this seems not to work with flat objects. Also make sure Bending Constraints are enabled. Otherwise the paper can just fold along the edges with no resistance. Also increase the Margin so that collisions will be detected correctly. Be aware however that this will make your paper float slightly above the ground.

This is what the resulting paper looks like when falling onto the ground:



One thing that is still missing however is air drag. Maybe you can simulate that using a Force field constraint actuator but I have not found out how they work yet.